3.854 \(\int \frac{1+x^6}{x \left (1-x^6\right )} \, dx\)

Optimal. Leaf size=15 \[ \log (x)-\frac{1}{3} \log \left (1-x^6\right ) \]

[Out]

Log[x] - Log[1 - x^6]/3

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Rubi [A]  time = 0.0497734, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \log (x)-\frac{1}{3} \log \left (1-x^6\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^6)/(x*(1 - x^6)),x]

[Out]

Log[x] - Log[1 - x^6]/3

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Rubi in Sympy [A]  time = 7.85375, size = 14, normalized size = 0.93 \[ \frac{\log{\left (x^{6} \right )}}{6} - \frac{\log{\left (- x^{6} + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**6+1)/x/(-x**6+1),x)

[Out]

log(x**6)/6 - log(-x**6 + 1)/3

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Mathematica [A]  time = 0.00962285, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{3} \log \left (1-x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^6)/(x*(1 - x^6)),x]

[Out]

Log[x] - Log[1 - x^6]/3

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Maple [B]  time = 0.014, size = 36, normalized size = 2.4 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}}-{\frac{\ln \left ( -1+x \right ) }{3}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}}-{\frac{\ln \left ( 1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^6+1)/x/(-x^6+1),x)

[Out]

ln(x)-1/3*ln(x^2+x+1)-1/3*ln(-1+x)-1/3*ln(x^2-x+1)-1/3*ln(1+x)

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Maxima [A]  time = 1.41852, size = 20, normalized size = 1.33 \[ -\frac{1}{3} \, \log \left (x^{6} - 1\right ) + \frac{1}{6} \, \log \left (x^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^6 + 1)/((x^6 - 1)*x),x, algorithm="maxima")

[Out]

-1/3*log(x^6 - 1) + 1/6*log(x^6)

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Fricas [A]  time = 0.217254, size = 15, normalized size = 1. \[ -\frac{1}{3} \, \log \left (x^{6} - 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^6 + 1)/((x^6 - 1)*x),x, algorithm="fricas")

[Out]

-1/3*log(x^6 - 1) + log(x)

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Sympy [A]  time = 0.127994, size = 10, normalized size = 0.67 \[ \log{\left (x \right )} - \frac{\log{\left (x^{6} - 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**6+1)/x/(-x**6+1),x)

[Out]

log(x) - log(x**6 - 1)/3

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GIAC/XCAS [A]  time = 0.210549, size = 22, normalized size = 1.47 \[ \frac{1}{6} \,{\rm ln}\left (x^{6}\right ) - \frac{1}{3} \,{\rm ln}\left ({\left | x^{6} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^6 + 1)/((x^6 - 1)*x),x, algorithm="giac")

[Out]

1/6*ln(x^6) - 1/3*ln(abs(x^6 - 1))